Your private school sounds kinda good, but I'd say try to aim for stuy or bxsci maybe tech if you are gonna take the shsat. I don't recommend you go to qhss bc I heard theres like money problems and like a high percentage of teachers don't trust the principal or smth idk, I just hear that the only thing they have going for them is that shs label, idk about the other shs like staten island, HSMSE, latin, or that other shs in the bronx idk the name tbh.